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Math Help - simple question on R module homomorphism

  1. #1
    Newbie dangkhoa's Avatar
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    simple question on R module homomorphism

    Let f: M ---> N be a R module homomorphism
    P and Q are submodules of M and N respectively.
    Show that:
    (i) f(P) ={ f(p): p belongs to P} is a submodule of N
    (ii) f^(-1)(Q) ={ m in M: f(m) belongs to Q} is a submodule of M
    (iii) What are f(M) and f^(-1)(0)?

    I have already proved part (i) and (ii)
    My problem is part (iii). I don't know how to do it, it looks easy but I think it is a trick question. So, I have to bother you to help me on part(iii)

    Thank you in advanced
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  2. #2
    Senior Member Shanks's Avatar
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    f(M) is the image of f. f^{-1}\{0\} is the kernal of f.
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  3. #3
    Super Member Gamma's Avatar
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    Yeah, I think you are just supposed to note that those are two very special cases of what you proved in i) and ii). If f(M)=N f is surjective (onto) and if f^{-1}(0)=\{0\} then f is injective (one to one).
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